Page 1
Math 151-copyright Joe Kahlig, 09B
1. (6 points) Given m = h2, 7i and n = h1, 10i. Find the vector projection of n onto m
2. (6 points) Find a vector equation of the line thru the points (1, 7) and (3, 12)
3. (15 points) A pilot wishes to set a course so that his ground speed is northeast(N45o E) at 180 mph. the
wind is blowing in the direction of S30o E at 40 mph. What course (speed and bearing) should the pilot
set in order to achieve his desired ground speed?
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Page 2
Math 151-copyright Joe Kahlig, 09B
4. (16 points) Find the derivatives for the following. Assume that g(x) is a differentiable function. DO NOT
SIMPLIFY YOUR ANSWER.
√
1
5
(a) f (x) = x2 x2 + 4
5x
(b) h(x) = (x4 + 3x + 7)g(x)
5. (9 points) Find the equation of the tangent line at x = 2 for the function f (x) =
x3 + 1
.
x+8
6. (8 points) Determine the values of x where the function is not continuous. For each of these numbers
state whether f is continuous from the right, or from the left, or neither.
6
5
4
3
2
1
−7 −6 −5 −4 −3 −2 −1
−1 1
2
3
4
5
6
7
8
9
10
11 12
−2
−3
−4
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Page 3
Math 151-copyright Joe Kahlig, 09B
7. (8 points) Let r(t) =< t3 + t + 1, t2 − 9 >. Find a tangent vector(s) at the point (11, −5).
8. (20 points) Find the exact values of the following limits. (continued on the next page)
2x2 − 7x + 3
=
x→3
x2 − 9
(a) lim
(b)
x4 + 3x
=
x→−∞ 4 − x2
lim
√
x + 2x2 + 1
(c) lim
=
x→−∞
4x + 1
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Page 4
Math 151-copyright Joe Kahlig, 09B
(d) lim
x→2
1
4
− 2
x−2 x −4
=
9. (6 points) For what value(s) of c and m that will make the function f (x) be differentiable everywhere. If
this can not be done, then explain why. Fully justify your answers.
x2
for x < 3
f (x) =
cx + m for x ≥ 3
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Page 5
Math 151-copyright Joe Kahlig, 09B
10. (6 points) Use the definition of the derivative to show that the derivative of f (x) = x2 + 5x is f ′ (x) = 2x + 5.
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